In Section REF ,
we briefly describe the algorithm of Schrijver [1]} so that its description is compatible with our notations.
Then we analyze the running time of this algorithm in a fine-grained manner with respect to
structural parameters of \(G^\textsf {p}\)  as well as the complexity of the graph. Using a suitable data structure, we can implement the
algorithm of of Schrijver [1]} in \(\textsf {poly}(\chi )\cdot n\)  time, where \(\chi \)  is a structural parameter we used in the analysis. We will show that the structural parameter \(\chi \)  has value \(2^{O(k)}\)  in our case.
To obtain a good bound on \(\chi \) , we slightly modify \(G\)  into another planar graph \(G^{\prime }\)  in Section REF . Then we analyze an upper bound on the structural parameter
in Section REF .
By combining these arguments, we can conclude that the Planar Disjoint Paths can be solved in \(2^{O(k^2)}n\)  time.
